High-energy cosmic rays passing through the Earth's
atmosphere produce extensive showers whose charges emit radio
frequency pulses. Despite the low density of the Earth's
atmosphere, this emission should be affected by the air
refractive index because the bulk of the shower particles move
roughly at the speed of radio waves, so that the retarded
altitude of emission, the relativistic boost, and the emission
pattern are modified. In this paper we consider the contribution
of the boosted Coulomb and the Čerenkov fields and
calculate analytically the spectrum using a very simplified model
in order to highlight the main properties. We find that typically
the lower half of the shower charge energy distribution produces
a boosted Coulomb field, of amplitude comparable to the levels
measured and to those calculated previously for synchrotron
emission. Higher energy particles instead produce a
Čerenkov-like field, whose amplitude may be smaller because
both the negative charge excess and the separation between
charges of opposite signs are small at these energies.

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